Confinement and the infrared behaviour of the gluon propagator

Abstract

We investigate the infrared behaviour of the gluon propagator in Quantum Chromo- dynamics (QCD). A natural framework for such a non-perturbative study is the complex of Schwinger-Dyson equations (SDE).The possible infrared behaviour of the gluon, found by self-consistently solving the approximate boson SDE, is studied analytically. We find that only an infrared enhanced gluon propagator, as singular as 1/p(^4) as p(^2) → 0, is consistent and demonstrate why softer solutions, that others have found, are not allowed. Reassuringly the consistent, enhanced infrared behaviour is indicative of the confinement of quarks and gluons, implying, for example, area-law behaviour of the Wilson loop operator and forbidding a Kāllen-Lehmann spectral representation of both quark and gluon propagators. We then briefly consider the implications of these results for models of the pomeron. The enhancement of the gluon propagator does however introduce infrared divergences in the SDE and these need to be regularised. So far model forms of the enhanced gluon propagator have been used in studies of dynamical chiral symmetry breaking and hadron phenomenology. Though very encouraging results have been obtained, one might hope to use the gluon propagator obtained directly from non-perturbative QCD to calculate hadron observables. We therefore attempt to eliminate the infrared divergences in the SDEs in a self- consistent way, entirely within the context of the calculational scheme. To do this we introduce an infrared regulator λ in the truncated gluon SDE in quenched QCD. We find that this regulator is indeed determined by the equation and bounded by the QCD-scale Aqcd- Thus it is possible to perform the regularisation within the SDEs. However, we have not been able to choose λ < Aqcd.